Rogers-Ramanujan type identities at $\Lambda_0$ from perfect crystals of exceptional quantum affine algebras

Abstract

We derive Rogers--Ramanujan type partition identities at the fundamental weight $\Lambda_0$ for the exceptional affine types $G_2^{(1)}$, $D_4^{(3)}$, $F_4^{(1)}$, $E_6^{(2)}$, $E_6^{(1)}$, $E_7^{(1)}$ and $E_8^{(1)}$. Our starting point is the Dousse--Konan reformulation of the $(\mathrm{KMN})^2$ crystal character formula, applied to the level-one perfect crystal $B=B(\theta)\sqcup B(0)$ of Benkart--Frenkel--Kang--Lee with ground element $\phi\in B(0)$. This realizes the normalized character $e^{-\Lambda_0}\mathrm{ch} L(\Lambda_0)$ as generating functions of grounded $B$-colored partitions governed locally by the crystal energy. After principal specialization, we obtain a colored partition model subject to explicit difference, congruence, and initial conditions. On the product side, under the same specialization, the Weyl--Kac character formula yields an explicit Euler-type product, equivalently the generating function for partitions with parts in a concrete allowed set. Comparing the two specializations gives coefficientwise equalities of generating functions. A key computational feature is that the difference matrix can be produced from the crystal data without explicitly computing the energy function. For each type we tabulate the congruence data, forbidden initial parts, and the full difference matrix, and we provide reproducible coefficient checks.